用粒子流匹配方法求解高维平均场博弈，突破计算瓶颈。
High-dimensional Mean-Field Games by Particle-based Flow Matching
- 基于粒子的流匹配方法，无须模拟直接学习轨迹速度
- 理论证明收敛速度达亚线性，凸性条件下可实现指数收敛
- 适用于非势能型博弈和高维生成问题，适合大规模优化场景
平均场博弈（MFG）研究连续交互代理系统的纳什均衡，可表述为最优控制问题的不动点。该框架广泛应用于最优传输（OT）与生成模型等领域。然而，高维MFG求解仍面临重大计算与分析挑战。本文提出一种基于粒子的深度流匹配（FM）方法，用于解决高维MFG计算问题。在每轮近端不动点迭代中，利用一阶信息更新粒子，并训练流神经网络以无模拟方式匹配样本轨迹的速度。理论上，在最优控制设定下，我们证明该方案可亚线性收敛至稳定点；在额外凸性假设下，收敛速度可提升至线性（指数级）。证明中通过流匹配将拉格朗日坐标（粒子基）转换为欧拉坐标（密度基），并得到两种形式在欧拉解足够光滑时的等价性结果。实验表明，该方法在非势能型MFG及通过松弛终端代价公式建模的高维OT问题上表现优异。
原文摘要 · Abstract (English)
Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents, which can be formulated as the fixed-point of optimal control problems. They provide a unified framework for a variety of applications, including optimal transport (OT) and generative models. Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles. In this work, we propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFG computation. In each iteration of our proximal fixed-point scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories in a simulation-free manner. Theoretically, in the optimal control setting, we prove that our scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions. Our proof uses FM to induce an Eulerian coordinate (density-based) from a Lagrangian one (particle-based), and this also leads to certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficiently regular. Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT problems cast as MFGs through a relaxed terminal-cost formulation.
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